Fuzzy bang-bang relay controller for satellite attitude ... paper/c14… · other hand, a bang-bang...

Fuzzy Sets and Systems 161 (2010) 2104–2125www.elsevier.com/locate/fss

Fuzzy bang-bang relay controller for satellite attitudecontrol system

Farrukh Nagi∗, S.K. Ahmed, A.A. Zainul Abidin, F.H. NordinUniversity Tenaga Nasional, Department of Mechanical Engineering, Km 7, Jalan Kajang Puchong, Kajang 43009, Selangor, Malaysia

Received 5 July 2007; received in revised form 4 May 2009; accepted 8 December 2009Available online 21 December 2009

Abstract

Two level bang-bang controllers are generally used in conjunction with the thrust reaction actuator for spacecraft/satellite attitudecontrol. These controllers are fast acting and dispense time dependent; full or no thrust-power to control the satellite attitude inminimum time. A minimum time-fuel attitude control system extends the life of a satellite and is the main focus of this paper.Fuzzy controllers are favored for satellite control due to their simplicity and good performance in terms of fuel saving, absorbingnon-linearities and uncertainties of the plant. A fuzzy controller requires a soft fuzzy engine, and a hardware relay to accomplishbang-bang control action. The work in this paper describes a new type of fuzzy controller in which the hardware relay action isconfigured in the soft fuzzy engine. The new configuration provides fuzzy decision-making flexibility at the inputs with relay liketwo-level bang-bang output. The new fuzzy controller is simulated on a three-axis satellite attitude control platform and comparedwith conventional a fuzzy controller, sliding mode controller and linear quadratic regulator. The result shows that the proposedcontroller has minimum-time response compared to other controllers. Inherent chattering associated with a two-level bang-bangcontroller produces undesirable low amplitude frequency limit cycles. The chattering can be easily stopped in the proposed fuzzybang-bang relay controller, hence adding multi-functionality to its simple design.© 2009 Elsevier B.V. All rights reserved.

Keywords: Bang-bang; Slide mode control; Fuzzy logic control; State dependent Riccati equation

1. Introduction

Minimum response time of a control system complements energy-saving measures especially in embedded controlsystems. A satellite attitude control system is one such example where fuel saving is highly desirable. Likewise,remotely controlled submersibles, deep space exploration probes can also benefit from such measures but to a lesserextent than communication satellites due to their high launching cost. Minimum response time also ensures that satelliteorientation error can be efficiently removed without degrading the performance of the satellite.

Actuators such as reaction wheels, magnetic torques, or reaction thrusters control satellite attitude. The attitude controlis described as fast or slow depending upon the slew angle ranges of the mission. Fast attitude control systems employthrusters, which gives large control torques to counter large slew angle disturbances arising due to Earth’s gravitation,airflow, magnetic fields, and solar winds. Thrusters can work as proportional (analog) or two-level switching bang-bang

∗ Corresponding author.E-mail address: [email protected] (F. Nagi).

0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2009.12.004

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F. Nagi et al. / Fuzzy Sets and Systems 161 (2010) 2104–2125 2105

controls. Proportional controllers are precise but problematic and costlier than bang-bang controllers. The variable jetnozzles of proportional controllers easily get clogged with dirt, debris and ice, resulting in failed operation. On theother hand, a bang-bang controller uses large and constant area nozzles. Full discharging bang-bang thrusters can openthe spring-loaded flaps covering the nozzles from the elements [12].

In a bang-bang control system the valves can be reliably operated to stay open for as little as a few milliseconds.The full opening of the valves for a finite time changes the discrete angular velocity with each actuation. As a resultit is not possible to get zero residual angular velocity. To prevent opposing thrusters fighting each other, a deadbandis introduced between the on–off control, and the controller is shut down in this deadband region. As a result thecontrolled system coasts to the equilibrium point (origin) with reduced velocity or increased damping. This generateslow frequency (and amplitude) limit-cycles, and hence dissipates the thruster force.

Simple hardware relays or hard limiter programming functions are used for implementation of the bang-bang controlsystem. In the absence of a deadband in bang-bang control, hysteresis is commonly used to compensate for the deadbandaction. Hysteresis provides a neat solution, but lacks robustness as it is fixed or chosen to optimize performance forone particular nominal plant operating condition.

One solution is to use flexible hysteresis for bang-bang control action. Mazari and Mekri [2] show how the flexiblefuzzy hysteresis band (HB) is used to optimize the pulse width modulator (PWM) for reduction of harmonic currentpollution. Nowadays, digital processors are readily available to implement software based fuzzy and neural networkcontrollers. Fuzzy controllers are flexible, simple to build and provide robustness to the bang-bang controller. In adetailed literature review Lee [10] reported a wide range of non-linear systems controlled by fuzzy logic. Berenji etal. [3] proposed a fuzzy control scheme for attitude stabilization of a space shuttle. Bang-bang fuzzy logic controllershave been developed in the past. A bang-bang fuzzy controller developed by Chiang and Jang [16], made its debut inCassini spacecraft’s deep space exploration project. The controller proves its superiority over the conventional bang-bang controller. Other applications include minimum-time fuzzy satellite attitude controller [3], crane hoisting/loweringoperation [13]. However, these applications are not well suited to the standard linear control design methodologies. Theapplications described in [3,13] use fuzzy bang-bang control. The defuzzified outputs in these works were based oncenter of gravity (COG), center of area (COA) or centroid methods. These methods yield crisp analog outputs, whichare converted to two level-control using hard limiting devices.

The idea of fuzzy bang-bang relay is not new. Kendal and Zhang [1] and Keichert and Mamdani [21] first pointed outthat fuzzy controllers are identical to multilevel relays, when using mean of maxima (MOM) defuzzification technique.Fuzzy relay in power control application was first presented by Panda and Mishra [8], where the relay was used inconjecture with fuzzy rules. Hard limiter was used in this work to convert the defuzzified output to two level controls.

New controllers are not acceptable to the control community, unless their stability is proven using establishedtechniques. In the case of a fuzzy controller, the heuristic approach of fuzzy rules results in partitioning of the decisionspace (phase plane) into two semi-planes by a sliding (switching) line. Similarity between a fuzzy bang-bang controllerand a sliding mode controller (SMC) can be used to redefine the diagonal form of an FLC in terms of an SMC, withboundary limits, to verify the stability of the proposed bang-bang controller [5,15]. SMC is a robust control method[19] and its stability can be proven using Lyapunov’s direct method. So in lieu of SMC, the fuzzy bang-bang controlstability can be easily established.

SMC requires a non-linear representation of the system to design the controller. It has an undesirable characteristicof producing high frequency control signals, which can shake the system vigorously. This drawback could be removedby using fuzzy logic controllers [14]. A fuzzy logic controller results in a highly non-linear control scheme [22] inwhich thrusters are active for a given duration. The controller has to evaluate the time and sign of the switch of eachthruster. Three MISO fuzzy controllers are used for attitude stabilization of a small satellite in a low earth orbit, provingthat fuzzy controller solves the control constraint problem by choosing the appropriate duration and polarity of thethrusters.

The minimum-time control is highly desirable for bang-bang controller design, especially for satellite attitude control[20]. Minimum control discussion is not complete without mentioning the celebrated Riccati equation. LQR methodsare applicable to linear systems; however, state-dependent Riccati equation (SDRE) [6] is a promising technique forcontrolling non-linear systems, such as the six-degree of freedom satellite attitude control system considered in thispaper. In SDRE, the state matrix and control law are updated and used as linear approximations at every sample time.The advantage of SDRE is its efficiency in converging to a solution and is a primary source of spacecraft trackingcontrol systems [7].



2106 F. Nagi et al. / Fuzzy Sets and Systems 161 (2010) 2104–2125

In this paper a new configuration of fuzzy controller is proposed. This controller directly produces a two-level bang-bang crisp output. The inputs to the controller are configured on standard fuzzy sets based on Mamdani inference. Here,the Takagi–Sugeno–Kang (TSK) inference is not suitable as the weighted average of rules gives a singleton output,which is linear or has a constant level. However, the bang-bang controller has only two output levels. The controller doesnot require any further saturation or hard limiting devices. The fuzzy rules’ consequents parts have only two linguisticvalues, while the premise parts are chosen freely. The fuzzy bang-bang controller, which works like a two-level relayand is referred to as a fuzzy bang-bang relay controller (FBBRC). Due to fixed two level bang-bang output the FBBRCdecision making capability is limited to its inputs only. Further, being software based the FBBRC is switched off oncethe desired angle is achieved within the specified deadband limit. The FBBRC is also computationally less demandingthan an LQR controller. In the next section, satellite kinematics and dynamics are defined and modeled in state spaceform. In Section 3 the proposed controller, FBBRC, is developed. An SDRE and SMC controllers are describedin Section 5 and Appendix B, respectively, followed by simulations and comparison of controllers’ performance inSection 6.

2. Three degree of freedom satellite state space model

Three degree of freedom (3-dof) rigid satellite model is presented in this section. The attitude motions about the threeprincipal axes are nearly uncoupled, and stabilization about each principal axis may be treated separately. Axes X B ,YB , and Z B define the satellite’s body axis frame, and the axis system is considered centered at the center of gravityas shown in Fig. 1. Thrusters are available to produce torques about each of the three principal axes. The physicalinterpretation of the Euler angles [18] for a micro-satellite platform is illustrated in Fig. 1. The roll (�), pitch (�), andyaw (�) angles are defined by successive rotations around the coordinate axes X B, YB, and Z B in the body fixed frame.For large angles and position control [7,4] of spacecraft, quaternion rotation is used. This is more suitable as it avoidssingularity functions and gimbal lock [9].

Euler angular moment (H ) [18] which performs satellite attitude-rotational motion in space is expressed in the formof a matrix as follows:

M = d H

dt+ [� × H ]

d H

dt= M − [� × H ] because H = I · �

I · d�

dt= M − [� × I · �]

d�

dt=

⎡⎢⎢⎣

·p·q·r

⎤⎥⎥⎦ = I −1 · M − [I −1 · �] × [I · �] (1)

B

B

B

Roll: , p

Pitch: , q

Yaw: , r

�

�

Y

X

Z

Fig. 1. Satellite reference and body coordinates.




where vector M , the applied moment (thrusters), is the input u, vector � is the angular rate and matrix I is the inertiaaround the body principal axes X B , YB , and Z B . These are given by

u = [Mx , My, Mz]T

� = [p, q, r ]T

I =

⎡⎢⎢⎣

Ixx 0 0

0 Iyy 0

0 0 Izz

⎤⎥⎥⎦ (2)

Solving (1) and (2)

⎡⎣ p

qr

⎤⎦ =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

Mx − q · Izz · r + r · Iyy · q

Ixx

My − r · Ixx · p + p · Izz · r

Iyy

Mz − p · Iyy · q + q · Izz · p

Izz

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(3)

The relationship between body rates �b = [pb, qb, rb]T of the satellite and angular rates � = [�, �, �]T is defined bythe following transformation [18]:

⎡⎣ �I

�I

�I

⎤⎦ =

⎡⎢⎢⎢⎣

1sin(�) sin(�)

cos(�)

sin(�) sin(�)

cos(�)0 cos(�) − sin(�)

0sin(�)

cos(�)

cos(�)

cos(�)

⎤⎥⎥⎥⎦ ·

⎡⎣ pb

qb

rb

⎤⎦

=

⎡⎢⎢⎢⎣

P + (q sin(�) + r cos(�)) sin(�)

cos(�)q cos(�) − r sin(�)q sin(�) + r cos(�)

cos(�)

⎤⎥⎥⎥⎦ (4)

The non-linear state model of the satellite can be derived by partial derivatives of the model states x = [pb, qb, rb, �I ,

�I , �I ]T

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

·pb·qb·rb·�I�I

�I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

= f (x, u) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Mx − q · Izz · r + r · Iyy · q

IxxMy − r · Ixx · p + p · Izz · r

IyyMz − p · Ixx · r + q · Iyy · p

Izz

p + (q sin(�) + r cos(�)) sin(�)

cos(�)q cos(�) − r sin(�)q sin(�) + r cos(�)

cos(�)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦




Table 1Satellite parameters.

Parameters Description Value

Ixx Moment of inertia (x-axis) 1.928 kg m2

Iyy Moment of inertia (y-axis) 1.928 kg m2

Izz Moment of inertia (z-axis) 4.953 kg m2

Thruster Input to satellite (Mx , My , Mz) 1 kg m2

�0 Initial roll Euler angle 0.362 rad�0 Initial pitch Euler angle 0.524 rad�0 Initial yaw Euler angle −0.262 radp Body pitch roll rate 0 rad/sq Body yaw rate 0 rad/sr Body roll rate 0 rad/s� Deadband ±10 mrad

The non-linear state space model can be found by taking the Jacobians A = Jacobian( f (x), x) andB = Jacobian( f (x), u):

A(x)=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0−r Izz + r Iyy

Ixx

−q Izz + q Iyy

Ixx0 0 0

−r Ixx + r Izz

Iyy0

−pIxx + pIzz

Iyy0 0 0

−q Iyy + q Ixx

Izz

−pIyy + pIxx

Izz0 0 0 0

1sin � sin �

cos �cos � sin �

cos �q cos � − r sin � sin �

cos �q sin � + r cos � + (q sin � + r cos �)(sin2 �)

cos2 �0

0 cos � − sin � −q sin � − r cos � 0 0

0sin �cos �

cos �cos �

q cos � − r sin �cos �

q sin � + r cos � sin �cos2 �

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

B =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1

Ixx0 0 0 0 0

01

Iyy0 0 0 0

0 01

Izz0 0 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(5)

where matrix A(x) and B(u) are Jacobian matrix of non-linear function f (x)with respect to state vector x = [p q r � ��]T and input vector u = [Mx My Mz 0 0 0]T , respectively. The satellite parameters in Table 1 are based on themodel referenced in Thongchet and Kuntanapreeda [20].

3. Fuzzy bang-bang relay controller

A bang-bang fuzzy relay controller is developed in this section. The controller is developed for only roll-axisand is identical for the other two axes. The controller takes advantage of largest of maxima defuzzification (LOM)technique to yield bang-bang output directly. For any fuzzy controller it is necessary to determine the ranges of itsstate, input and output variables. These should be a reasonable representation of all the situations the controller willface, and should yield to stability and optimality conditions. The following ranges are selected for simulation purposes:�(t) = [−3, 3] rad, �(t) = [−3, 3] rad/s and control signal ur = [−Mx , +Mx ].




Fig. 2. Three degree of freedom satellite model and fuzzy bang-bang relay controllers.

3.1. Linguistic description

The input and output variables of the fuzzy controllers are as shown in Fig. 2. The input and output parameters,partitions and spread of the controller membership functions are selected to match the dynamic response of the satellite.The inputs xi ∈ i , where i , i = 1, 2 is the universe of discourse of the two inputs. For linguistic input variable,x1 = “error angle”, the universe of discourse, 1 = [−3, 3] rad, represents the angle of perturbation angle from the




Table 2Fuzzy rules for FBBRC.

� h

LN SN Z SP LP

LN +Mx +Mx +Mx +Mx

SN +Mx +Mx +Mx −Mx

Z +Mx +Mx −Mx −Mx

SP +Mx −Mx −Mx −Mx

LP −Mx −Mx −Mx −Mx

Table 3Fuzzy rules for standard FLC.

� h

LN SN Z SP LP

LN PL* PL PL PS OFFSN PL PL PS OFF NSZ PL PS OFF NS NLSP PS OFF NS NL NLLP OFF NS NL NL NL

*PL, positive large; PS, positive small; NL, negative large.

zero reference and for linguistic input variable x2 =“error angle rate,” the universe of discourse is 2 = [−3, 3] rad/s.The output universe of discourse = [−Mx , +Mx ] represents the bang-bang output ∈ .

The set A ji defines the jth linguistic value of linguistic variable xi , which is defined over the universe of discourse

i . The control level of the system operation can be adequately defined for input x1 by the following linguistic values:

A ji=1 = [ A1

1 = L N , A21 = SN , A3

1 = Z , A41 = S P, A5

1 = L P]

Similar linguistic values are selected for input x2; i.e., A j2 ≡ A j

1. The set B j1 denotes the linguistic values for output

linguistic variable y1 and is defined by

B ji = [B1

1 → J2, B21 → J1]

where J1 and J2 are on/off-firing command for the thrusters.

3.2. Fuzzy rules

The fuzzy linguistic rules assembled in this work reset the angle � = 0◦. The rules are based on two input variables,each with five linguistic values. Thus there are at most 25 possible rules. The rules are described in matrix form inTables 2 and 3. In later sections it will be shown that the rules can be interpreted from sliding mode control. The maindiagonal entry in the rule Table 2 is not used. The rules-partitions are heuristically chosen to reset the beam smoothlyover the universe of discourse.

The rules matrix symmetry is no surprise and arises from the symmetry of the system dynamics. The decompositionof linguistic rules from the controller’s input side to the output is

�B j

ig(y) = min{�

A j1g

(x1), �A j

2g(x2)} (6)

The index g is for the number of rules used in the implication.




x1

Deg

ree

of m

embe

rshi

pSN

21A

~ =

31A

~= ZLN

11A

~ =

SP41A

~ =LP

51A

~ =

j1A

~μ

-2 -3

1

0.5

0

-1 10 2 3

Fig. 3. Membership functions of input � ⇒ x1 = “error angle” and linguistic values A j1 for both FBBRC and standard FLC.

Deg

ree

of m

embe

rshi

p

12A

~=LN 2

2A~

=SN

32A

~=Z

42A

~=SP

52A

~=LP

j2A

~µ

x2

1

0.5

0

-3 -2 -1 10 2 3

Fig. 4. Membership functions of input � ⇒ x2 = “error angle rate” and linguistic values A j2 for both FBBRC and standard FLC controller.

3.3. Fuzzy set membership functions

The input linguistic variables and values assigned to fuzzy set membership functions are shown in Figs. 3 and 4.Triangular shape membership functions are used in this work, which are sensitive to small changes in the vicinity of theircenters. A small change across the central membership function, A3

i , located at the origin, can produce abrupt switching.This effect is countered by asymmetrical (smaller) spread of A3

1. Center width narrow (CWN) membership functionwith the narrow width near the origin is used to prevent overshoot [17], Figs. 3 and 4. Adjusting A3

1 overlaps with theadjacent membership functions control its degree of sensitivity. The input state near the origin quickly contributes tothe switching of control command u between +ve and −ve half of universe of discourse, producing chattering. Theoverlapping of the central membership function A3

1 with the neighboring membership functions A21 and A4

1 providesinsensitivity—fuzzy hysteresis to bang-bang control action. The hysteresis controls the switching time delay betweenJ1 and J2.

Triangular membership functions in Figs. 3 and 4 are based on mathematical characteristics given in Table 4. Smoothtransition between adjacent membership functions is achieved with higher percentage of overlap, which is commonly setat 50%. The output membership function for standard FLC is shown in Fig. 5 and for FBBRC see Fig. 6. In FBBRC notethe absence of a central membership function at the origin of the output universe of discourse. This is also reflectedby the absence of diagonal rules in Table 2. Standard FLC uses the same input membership function as shown inFigs. 3 and 4.




Table 4Mathematical characterization of triangular membership functions.

Linguistic value Triangular membership functions

A1i

cL = −3 � A1i =B1

1(x) =

⎧⎨⎩

1, x = −3

max

{0, 1 + x − cL

−3

}, −2.9 ≤ x ≤ 0

A2i

cI L = −0.3 � A2i(x) =

⎧⎪⎪⎨⎪⎪⎩

max

{0, 1 + x − cI L

27

}, −3 ≤ x ≤ −0.4

max

{0, 1 + x − cI L

−0.3

}, −0.3 ≤ x ≤ 0

A3i

c = 0 � A3i(x) =

⎧⎪⎪⎨⎪⎪⎩

max

{0, 1 + x − c

3

}, −0.3 ≤ x ≤ 0

max

{0, 1 + x − c

−0.3

}, 0 ≤ x ≤ 0.3

A4i

cI R = 3 � A4i(x) =

⎧⎪⎪⎨⎪⎪⎩

max

{0, 1 + x − cI R

3

}, 0 ≤ x ≤ 0.3

max

{0, 1 + x − cI R

−2.7

}, 0.4 ≤ x ≤ 3

A5i

cR = 3 � A5i =B2

1(x) =

⎧⎨⎩

max

{0, 1 + x − cR

3

}, 0 ≤ x ≤ 2.9

1, x = 3

Fig. 5. Output membership functions of FLC, min–max aggregation and centroid.

3.4. Largest of maximum (LOM) aggregation

The selection of membership functions in Fig. 6a and LOM aggregation formulates the fuzzy bang-bang relaycontroller. Any input perturbation of the beam from the zero reference acts on the output membership functionsaccording to the rules matrix in Table 2.




j

iB~μ 1j

11

0.5

0

B~ = = J2

y- Mx + Mx

21B

~= J1

J1

Aggregation : min-max Defuzzification : LOM

Deg

ree

of

mem

bers

hip

-1

0

0

0

1

J2

ycrisp

Con

trol

ler

outp

ut u

- Mx +Mx

Fig. 6. (a) FBBBRC output y membership functions and linguistic values B j1 . (b) FBBRC two level bang-bang ycrisp output.

Let �overall be the membership function of the overall implied fuzzy rules g = 1, 2 . . . G. Then, it is obtained bytaking the maximum [10] as follows:

�overall (y) = maxg

{�B11g

(y), �B22g

(y), . . . , �B j

ig(y)} (7)

The defuzzified crisp output ycrisp is obtained as

sycrisp = arg sup{�overall (y)

}(8)

The supremum in (8) is the largest of maximum (LOM) value and will occur at the extremes of the output of theuniverse of discourse = [−Mx , Mx ]. The argument arg(sup(�)) returns ycrisp = [−Mx , Mx ]. The results of firingaction of membership functions J2 and J1 are shown in Fig. 6b.

4. Fuzzy sliding mode controller

The fuzzy rules described in Table 2 can be systematically constructed on the basis of sliding mode control andhitting condition described by Eq. (B.13) in Appendix B. Appendix B provides detailed derivation of the control lawand stability condition of the SMC and the Simulink simulation block diagram.

The rules in Table 2 can be deduced from Eq. (A.8). Multiplying it with s yields

ss = f (�; t)s + b(�; t)us + ��s (9)

For b > 0, if s < 0, then increasing u(+Mx ) will result in decreasing ss; and if s > 0, then decreasing u (−Mx ) willresult in decreasing ss. The control value u should be selected so that ss < 0 for 0 < s > 0. The slope of sliding lineis represented by �.

Considering s as � and s as·�, Mx = 1, u = [−1, +1], the fuzzy rules in Tables 2 and 3 and the membership functions

shown in Fig. 6a agree with the sliding mode condition, Appendix B.3.




5. State dependent Riccati equation (SDRE)

Linear quadratic regulator (LQR) is an established technique for satellite attitude control which uses the celebratedRiccati equation for the solution of minimum control signal. Linear quadratic regulator requires a linear system or anon-linear system linearized at an operating point for solving using Riccati equation. However, SDRE [16] can be usedfor non-linear systems such as satellite attitude control [17] and the non-linear state dependent matrix A(x) in (5) isupdated at every sample time. The non-linear state model (1)–(5) can be written in a linear state dependent coefficient(SDC) form of

·x = A(x) · x + B · u (10)

Consider satellite attitude as a non-linear regulator problem for minimizing, in infinite time, the performance criteriaJ given by

J = 1

2

∫ ∞

0[yT Q3×3(x)y3×1 + uT R3×3(x)u3×1] dt

The stability condition requires that Q, a state dependent weighting matrix is positive semi-definite while R, the controlweight matrix is positive definite. Solving the following Riccati equation for state dependent matrix P(x) > 0

AT (x)P + PA(x) − PB(x)R−1(x)BT (x)P + Q(x) = 0 (11)

Fig. 7. Euler angles reset comparison between three techniques from different initial positions.




Gives the non-linear state variable feedback control law u:

K = −R−1(x)BT (x)P(x)

u = −K · x (12)

Then u = [Mx My Mz 0 0 0]T in (12) is injected to the system model and new states x = [p q r � � �]T areobtained. The state dependent system matrix A(x) is updated and (12) is solved again and the cycle is repeated untilthe states reach the desired values. For satellite attitude reset the output u used here is

u = −K · x(p, q, r) + x(�, �, �) (13)

Simulation diagram for SDRE is provided in Appendix C.

6. Simulation response

The satellite model in simulation is based on Eqs. (1)–(5) and the parameters are given in Table 1 and Fig. 2. Thesimulated performance of the proposed controller is compared with other controllers.

The simulation response of the FBBRC, the centroid FLC, SMC and SDRE are shown in Fig. 7. Thruster strengthand cycle comparison of the controllers are shown in Fig. 8.

Both fuzzy controllers use the same input membership functions and initial conditions. However, the outputmembership functions are different.

The FBBRC has the output membership function as shown in Fig. 6 and standard FLC has centroid output membershipfunction shown in Fig. 5. Angle reset demonstrated in Fig. 7 shows that SRDE and FBBRC have close response butthe control effort u of SDRE are five time more in thruster strength and cycle in comparison to FBBRC.

An important feature of the controllers is shutting down of bang-bang action within the deadband, � = ±0.01 rad asshown in Fig. 2. This prevents the controllers from chattering, avoiding high frequency vibration and fuel expending atzero states. The output universe of discourse for both controllers is = [−1, +1]. These results are expected becausethe bang-bang control system is inherently a time optimal controller [19,11].

Fig. 8. Control signal u = [−Mx + Mx ] comparison among the various techniques for roll axis with deadband.




Any proposed control strategy should be supported by stability analysis for acceptance by the control system com-munity. Fuzzy bang-bang relay controller is no exception. As discussed earlier, conventional bang-bang control systemhas firm stability ground via sliding mode control, which uses Lyapunov like function to satisfy the stability criteria,which is discussed in Appendix B.1.

7. Conclusion

This paper proposes an integrated fuzzy bang-bang relay controller for satellite attitude control. Its capabilities weredemonstrated on simulated model and compared with other optimal control technique (LQR). The simulation model,representing a non-linear satellite model without uncertainties, was used for the development, stability, and optimalresponse. Fuzzy controllers are known for absorbing the non-linearity of the systems and, as the results show, it workswell for the real system.

The bang-bang control is inherently time-optimal, and this property is an important feature of the FBBRC. Compar-ison between the FBBRC and the standard FLC showed that FBBRC could reset the initial condition in optimal timeand with a smaller overshoot. FBBRC uses fewer output membership functions in comparison to the standard FLC andthus simplifying the unification process of the fuzzy rule matrix. In the FBBRC, the largest of maxima defuzzification(LOM) method yields direct bang-bang output from the fuzzy controller. The bang-bang output is similar to conven-tional hardware relay output. However, unlike fixed relay control/SMC inputs the FBBRC inputs are flexible to meetthe changing demands of the non-linear system dynamics. The stability and optimality of FBBRC were satisfied bythe SMC-Lyapunov criterion and optimal bang-bang control theory, respectively.

The FBBRC performs better than LQR optimal control using Riccati equation and SMC. For similar conditions theLQR method requires higher thruster power and more firing cycle than the FBBRC technique, thus reducing the sizeand fuel consumption.

Appendix A. Glossary

Symbols and abbreviations

A(x) Jacobian ( f (x), x)A fuzzy input setA j

i jth linguistic value of input linguistic variable xi

B(u) Jacobian ( f (x), u)B fuzzy output setB j

1 linguistic values for output linguistic variable yi

� deadband bounded condition for control gainCOA center of areaCOG center of gravityCWN center width narrowc center of triangular membership functioncL center leftcI L center inner leftcR center rightcI R center inner righte state error� reaching condition +ve constantFBBRC fuzzy bang-bang relay controllerFLC fuzzy logic controller (standard)




f (x) non-linear states function of 3 dof satellite systemg fuzzy rule index output universe of discourseGd derivative gainG number of fuzzy rulesHB hysteresis band·

H angular momentumi input indexI moment of inertiaJ performance indexj linguistic value indexJ control action boundsJ1 decomposition of linguistic rules �B2

1—on/off command

J2 decomposition of linguistic rules �B11—on/off command

K constant, sliding gain� slope of sliding lineLOM largest of maximaLQR linear quadratic regulatorM thruster momentum� decomposed value of fuzzy rule�overall decomposition of overall implied fuzzy rulesMISO multi input single outputMOM mean of maximaP state dependent matrixp body pitch rate� roll angle� yaw angleQ state dependent weight matrixq body yaw rateR control weight matrixRm dimensional states spanr body roll rates sliding switching lines rate of sliding lineSDC state differential coefficientSDRE state dependent Ricatti equationSMC sliding mode controller� pitch angle, output of interestu thrusters input vector� Euler angular rate vector�b body angular rate vectorx states vectorx LQR statesxi linguistic input variableX B body roll axisYB body pitch axisZ B body yaw axisi input universe of discoursey defuzzified outputyi linguistic output variableycrisp defuzzified crisp output




Appendix B. Sliding mode control (SMC)

A general 2nd order non-linear single-input–single-output (SISO) control system could be described [12] as

�(t) = f (�; t) + b(�; t)u (B.1)

where �(t) is the output of interest, u(t) is the scalar input, and � = [�,·�]T is state vector. In general, f (�; t) is not

precisely known, but upper bounded by a known continuous function of �. Similarly b(�; t) is not known, but is ofknown sign and is bounded by a known continuous function of x as

| f − f | ≤ F(�; t)

1

(�; t)≤ b

b≤ (�; t) (B.2)

where f and b are nominal values of f and b, respectively, without the function argument for brevity purpose.Comparing Eqs. (1) and (B.1), the system becomes

f (�; t) = −1

I�(t)

b(�; t)u = M

Iu(t) because b(�; t) = M

I(B.3)

where u(t) is a unit step input.The control problem is to get the state h to track hd = [�d �d ]T in minimum time and in the presence of imprecise

friction. The initial �d should be the following in view of finite control u:

�d (0) = �(0) (B.4)

The tracking error between the actual and desired state would be

e = � − � d=[e e]T (B.5)

A sliding-switching line s(�, t) in the second order state space R2 is defined such that e follows the line s(h, t) = 0.The sliding line s(h, t) is determined by

s(�, t) =(

d

dt+ �

)n−1

e (B.6)

Eq. can be expanded with binomial expansion and � is positive constant. For n = 2

s = e + �e because e = � and e = � (B.7)

Then from Eq. (B.1)

S = f (�; t) + b(�, t)u + �e (B.8)

B.1. SMC control law

Let ueq be the equivalent control law that will keep the states on the sliding trajectory, computed by s = 0 foru ≡ ueq, then from Eqs. (B.4), (B.5) and (B.7)

s = � + ��

s = � + �� (B.9)

Then from Eq. (B.8) with uncertainties

s|u=ueq = f (�; t) + b(�, t)ueq + �e = 0

Solving the above equation

ueq = b−1u (B.10)




x=[. . . , , ]

-α = -0.01 α = 0.01

s

λ

Thruster

xdot

UU(E)

U U(R,C)

MatrixMultiply

1

0.281

Moments1

Xe

Euler

p,q,r

Euler_rate

body_rate

A

inv B

states_dots

A.x

Dynamic Modelof

Three axis Sat ellite� ��

Fig. B1. Sliding mode controller described in Eq. (B.15).

where

u = [− f (�; t) − �e]

or

�e = − f (�; t) − u (B.11)

is the nominal control input in presence of uncertainties.The control input u to get the state h to track hd is then made to satisfy the Lyapunov-like function V = (1/2)s2,

if there exist � > 0 and by the following sliding condition [12]:

1

2

d

dts2(�, t) ≤ −�|s| (B.12)

which is reduced to the so-called sliding mode ‘reaching condition’ for Eq. (1)

s · sgn(s) ≤ −�|s|, � > 0 (B.13)

The control law that satisfies the sliding mode reaching conditions (Figs. B1 and B2) Eq. (B.13) can be obtained as

u = ueq + uS (B.14)

where

us = −K sgn(s) (B.15)




Fig. B2. Typical chattering control signal of sliding mode controller without deadband.

and

sgn(s) ={ +1 if s > 0

−1 if s < 0

Substituting Eqs. (B.1) and (B.8) in Eq. (B.13)

ss = s( f + bu + �e) ≤ −�|s|Note: Here we have dropped the function argument for brevity purpose. Then equivalently we can write

ss = sgn(s)( f + �e) + busgn(s) ≤ −�|s| (B.16)

To satisfy (B.16) control signal u is chosen as

u = 1

B(−A · x − � · x) − K sign(s) (B.17)

Substituting Eqs. (B.14) and (B.15) into Eq. (B.16)

ss = sgn(s)( f + �e) + b[ueq + b−1 K sgn(s)]sgn(s) ≤ −�|s|Substituting from Eqs. (B.10) and (B.11) in above, we get

ss = sgn(s) [ f + (− f − u)] + b [b−1u + b−1 K sgn(s)]sgn(s) ≤ −�|s|simplifying we get

sgn(s) ( f − f ) +[

b

b− 1

]usgn(s) − b

bK ≤ −�|s| (B.18)




φ roll

θ pitch

ψ yaw

roll rate

pitch rate

yaw rate

λ

0.281Thruster

UU(R,C)

U U(E)

MatrixMultiply

1

Moments1

Xe

Euler

p,q,r

Euler_rate

body_rate

A

inv B

A.x

Dynamic of

Three axis Sat x=[. . . , , ]

u

��

Fig. B3. Fuzzy logic controller formulated in Eq. (B.22).

Fig. B4. Bang-bang output of fuzzy controller without deadband.




Fig. C1. State dependent Riccati controller from Eqs. (11)–(13).

Then for upper bounds from Eq. (B.1) need

K ≥ [(F + �) + ( − 1)|u|] (B.19)

to satisfies the reaching or hitting condition.

Appendix B.2. Fuzzy sliding mode controller

Let K f uzz(x, x, �) be the absolute value of the control output of the FC. Then the controller term in (B.17) is

− K sign(s) = −K f uzz(x, x, k) · sign(s) (B.20)

Using the analogy of SMC, an important thing to observe here is that the strength of output u is constant as the distance|s| from the diagonal s = 0 increases, positive large—PL or negative large—NL. The bang-bang system requiresswitching sign(s) and K f uzz(|s|) equal to thruster output . Then the control law simplifies to

−k sign{s(x)} = −K f uzz(|s|)sign(s)

K f uzzy(|s|) = 0.281 Nm (Thruster) (B.21)

Putting (B.21) in (B.17) completes the control law (Figs. B3 and B4)

u = (�B)−1[−�·x −A · x)] − K f uzz(|s|) · sign(s) (B.22)

Appendix C. State dependent Riccati regulator

See Figs. C1 and C2.Controllers response describe in Appendices B and C (See Figs. C3–C5).




Fig. C2. Bang-bang output from SDRE controller without deadband.

Fig. C3. Roll angle comparison of SDRE, SMC and fuzzy techniques.




Fig. C4. Yaw angle comparison of three techniques.

Fig. C5. Pitch angle comparison of three techniques.

References

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